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Nine paradoxes with a statistical theme

A drill sergeant always yells at one of his trainees when she messes up. The drill sergeant notices that after he yells at her, her performance improves. Later it turns out that the trainee is deaf, blind, and has no other way of actually noticing that drill sergeant is yelling at her. Ignoring the effect of practice, why might the trainee’s performance have improved anyway?

You have 100 pounds of Martian potatoes, which are 99 percent water by weight. You let them dehydrate until they’re 98 percent water by weight. How much do they weigh now and why?

Imagine that your parents had rolled a six-sided die to decide how many children to have. What did they most likely roll and why?

You have access to planes that have returned from military missions and the distribution of the bullet “wounds” on the planes. Which areas should you recommend to have extra armor?

Why would few people choose to play in a lottery with a small but actual probability of success with an infinite monetary expected value?

Do most people have have the same, more, or fewer friends than their friends have on average and why?

Hypothetically, say that 80% of people dream in color, and 68% of sexual partners have the same (concordant) coloring of their dreams. If you dream in color, what’s the probability that your partner will too?

Are we biased to think that cars in the lanes next to us are going faster or slower than they really are and why?

Why is the expression “the smallest positive integer not nameable in under eleven words” paradoxical?

Answers

regression to the mean — the screwup is likely a random deviation below the trainee’s average, which will tend to improve on the subsequent iteration just due to random chance, regardless of any action by the drill sergeant (more here)

50 pounds, since the percentage of non-water by weight has doubled, so the overall weight must have halved (more here)

they most likely rolled a six, because there’s a higher chance of you existing to observe the event in that case (more here)

the areas with no damage, because of selection effects — planes that fell likely suffered an attack in a place that was untouched on those that survived (more here)

fewer, because sampling bias suggests that people with greater numbers of friends have an increased likelihood of being observed among one’s own friends (more here)

80%. Since basic probability theory tells us that 0.8 * 0.8 + 0.2 * 0.2 = 0.68, so we know that the probability of dreaming is color is independent of that of your sexual partner. Therefore, the probability that your partner dreams is color is independent of yours and is simply the base rate. Some people think 68%, perhaps because they are getting wrapped up in the causal story. (more here)

we are biased to think they are going faster, likely because because more time is generally spent being overtaken by other vehicles than is spent in overtaking them (more here)

there are finitely many words, so there are finitely many numbers that can be defined in under eleven words, so there must be such an integer, but since this expression itself is under eleven words, there cannot be any such integer (more here; resolved by assigning priority to the naming process either within or outside of the expression)